Editing Heat equation (section)

== Interpretation ==
Informally, the Laplacian operator {{math|∆}} gives the difference between the average value of a function in the neighborhood of a point, and its value at that point.  Thus, if {{mvar|u}} is the temperature, {{math|∆u}} conveys if (and by how much) the material surrounding each point is hotter or colder, on the average, than the material at that point.

By the [[second law of thermodynamics]], heat will flow from hotter bodies to adjacent colder bodies, in proportion to the difference of temperature and of the [[thermal conductivity]] of the material between them. When heat flows into (respectively, out of) a material, its temperature increases (respectively, decreases), in proportion to the amount of heat divided by the amount ([[mass]]) of material, with a [[proportionality (mathematics)|proportionality factor]] called the [[specific heat capacity]] of the material.

By the combination of these observations, the heat equation says the rate <math>\dot u</math> at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. The coefficient {{math|α}} in the equation takes into account the thermal conductivity, specific heat, and [[density]] of the material.

=== Interpretation of the equation ===
The first half of the above physical thinking can be put into a mathematical form. The key is that, for any fixed {{mvar|x}}, one has
: <math>\begin{align}
u_{(x)}(0)&=u(x)\\
u_{(x)}'(0)&=0\\
u_{(x)}''(0)&=\frac{1}{n}\Delta u(x)
\end{align}</math>
where {{math|''u''<sub>(''x'')</sub>(''r'')}} is the single-variable function denoting the ''average value'' of {{mvar|u}} over the surface of the sphere of radius {{mvar|r}} centered at {{mvar|x}}; it can be defined by
: <math>u_{(x)}(r)=\frac{1}{\omega_{n-1}r^{n-1}}\int_{\{y:|x-y|=r\}}u\,d\mathcal{H}^{n-1},</math>
in which {{math|ω<sub>''n'' − 1</sub>}} denotes the surface area of the unit ball in {{mvar|n}}-dimensional Euclidean space. This formalizes the above statement that the value of {{math|∆''u''}} at a point {{mvar|x}} measures the difference between the value of {{math|''u''(''x'')}} and the value of {{mvar|u}} at points nearby to {{mvar|x}}, in the sense that the latter is encoded by the values of {{math|''u''<sub>(''x'')</sub>(''r'')}} for small positive values of {{mvar|r}}.

Following this observation, one may interpret the heat equation as imposing an ''infinitesimal averaging'' of a function. Given a solution of the heat equation, the value of {{math|''u''(''x'', ''t'' + τ)}} for a small positive value of {{math|τ}} may be approximated as {{math|{{sfrac|1|2''n''}}}} times the average value of the function {{math|''u''(⋅, ''t'')}} over a sphere of very small radius centered at {{mvar|x}}.

=== Character of the solutions ===
[[Image:Heatequation exampleB.gif|right|frame|Solution of a 1D heat partial differential equation. The temperature (<math>u</math>) is initially distributed over a one-dimensional, one-unit-long interval (''x''&nbsp;=&nbsp;[0,1]) with insulated endpoints. The distribution approaches equilibrium over time.]]

[[File:Heat Transfer.gif|thumb|The behavior of temperature when the sides of a 1D rod are at fixed temperatures (in this case, 0.8 and 0 with initial Gaussian distribution). The temperature approaches a linear function because that is the stable solution of the equation: wherever temperature has a nonzero second spatial derivative, the time derivative is nonzero as well.]]

The heat equation implies that peaks ([[local maximum|local maxima]]) of <math>u</math> will be gradually eroded down, while depressions ([[local minimum|local minima]]) will be filled in.  The value at some point will remain stable only as long as it is equal to the average value in its immediate surroundings.  In particular, if the values in a neighborhood are very close to a linear function <math>A x + B y + C z + D</math>, then the value at the center of that neighborhood will not be changing at that time (that is, the derivative <math>\dot u</math> will be zero).

A more subtle consequence is the [[maximum principle]], that says that the maximum value of <math>u</math> in any region <math>R</math> of the medium will not exceed the maximum value that previously occurred in <math>R</math>, unless it is on the boundary of <math>R</math>. That is, the maximum temperature in a region <math>R</math> can increase only if heat comes in from outside <math>R</math>. This is a property of [[parabolic partial differential equation]]s and is not difficult to prove mathematically (see below).

Another interesting property is that even if <math>u</math> initially has a sharp jump (discontinuity) of value across some surface inside the medium, the jump is immediately smoothed out by a momentary, infinitesimally short but infinitely large rate of flow of heat through that surface. For example, if two isolated bodies, initially at uniform but different temperatures <math>u_0</math> and  <math>u_1</math>, are made to touch each other, the temperature at the point of contact will immediately assume some intermediate value, and a zone will develop around that point where <math>u</math> will gradually vary between <math>u_0</math> and <math>u_1</math>.

If a certain amount of heat is suddenly applied to a point in the medium, it will spread out in all directions in the form of a [[diffusion wave]].  Unlike the [[mechanical wave|elastic]] and [[electromagnetic wave]]s, the speed of a diffusion wave drops with time: as it spreads over a larger region, the temperature gradient decreases, and therefore the heat flow decreases too.